The high-energy behavior of scattering processes in perturbative quantum gravity of a very light scalar by a heavy scalar [1]},[2]},[3]} is considered in the eikonal and next-to-eikonal approximation [4]} for the large impact parameter and the consequent small scattering angle limit. These approximations schemes notably go beyond the finite order perturbation theory [5]},[6]}, allowing to sum the logarithmic divergent amplitudes in soft limit which have ladder like structure [7]},[8]},[9]}. The general idea of the eikonal limit is that at very high-energies the de Broglie wave-length of the incident particle of a scattering process is small compared with the spatial variations in the target, so that the “geometrical optics"-limit might be reasonable as a first approximation in which it is possible to recover the classical result of the exponentiation of the leading phase [10]}. That phase depends upon the mass dimension of the coupling constant which is for gravitational interactions inverse of squared energy [11]} and it is a fundamental infrared dominated physical quantity [12]}. By using and developing scattering amplitudes techniques [13]} more kinematic regimes and observables have been explored and also in generalized theories with gravity compared with classical gravitational scattering of massive objects untill two loop order [14]}. However, the approach of scattering amplitudes in perturbative quantum gravity to be compared with classical dynamics for the post Minkowskian expansion needs an expansion in the small scattering angle [15]} in which the amplitude can be resummed by exponentiating a suitably defined eikonal phase. As we will see in our analysis, there are subtleties about which perturbative diagrams do exponentiate and possibly, by the resummation approach used in this paper, the extension to two loops could be afforded in order to compare with the results from the scattering amplitudes. The extraction of classical results has been also studied in [16]}, [17]} and the eikonalization has been applied in that context [18]}.
In the reference [19]} a detailed analysis of the expansion around the eikonal limit directly in perturbation theory for the above mentioned scattering is provided and the diagrammatic self-consistency of this expansion is verified with the derivation of the next-to-leading corrections in arbitrary dimensions. The first derivation of the eikonal gravitational phase for the process in question can be found in [20]}. The next-to-leading correction has been also studied intensively [21]},[22]},[23]},[24]},[25]}. In our article we provide by the functional methods of quantum field theory the summation of diagrams corresponding to the treatment of the leading and the next-to-leading corrections of the high-energy scattering amplitude of the considered process. In the framework of the functional integral, the Green function representation in an external field proposed by Fradkin in [26]}, [27]} allows to perform calculations in a compact form with easy combinatorial counting based on the Wick’s contractions in the summations of ladder diagrams. Two nucleons scattering in quantum gravity was also analyzed by the so called Fradkin’s modified perturbation theory [28]}, however, our approach is based on the Feynman path integral and takes into account in the sum of the considered diagrams the full Feynman denominators like \((p+\sum k_i)^2-m^2 +i\epsilon \) , by expanding in the correlation terms \(k_i\cdot k_j\)  for the internal momenta. The quantum gravity regime is the appropriate to discuss the ultrahigh energy gravitational scattering in the eikonal approximation because it is characterized by the large impact parameter where the massless fields dominate. They cannot be strings modes because in [29]} has been proved that when the momentum transfers reach the string scale strings effects appear to be subdominant to higher loop gravitational processes approximated via eikonal. The high-energy scattering of massless closed strings from a stack of a given number of D-branes in Minkowski space is characterized also by the dominance of gravity when the impact parameter is large with respect to the Schwarzschild radius associated with the center of mass energy [21]}. The eikonalization procedure of [21]} is based on the use of Schwinger proper time, which in that context is expressed in terms of the strings and D-brane parameters in perturbation theory. In our approach the use of Schwinger proper time will allow an efficient eikonalization procedure with a suitable extension at next-to-eikonal approximation.
